New a Priori Estimates for Semistable Solutions of Semilinear Elliptic Equations

Abstract

We consider the semilinear elliptic equation -L u = f(u) in a general smooth bounded domain ${\Omega } \subset R^{n}$ with zero Dirichlet boundary condition, where L is a uniformly elliptic operator and f is a C 2 positive, nondecreasing and convex function in $[0,\infty )$ such that $\frac {f(t)}{t}\rightarrow \infty $ as $t\rightarrow \infty $ . We prove that if u is a positive semistable solution then for every 0 ≤ β < 1 we have $$\left|\left|f(u){{\int}_{0}^{u}}f(t)f^{\prime\prime}(t)~e^{2{\beta{\int}_{0}^{t}}\sqrt{\frac{f^{\prime\prime}(s)}{f(s)}}ds}~dt \right|\right|_{L^{1}({\Omega})}\leq C_{\beta}<\infty, $$ where C β is a constant independent of u. As we shall see, a large number of results in the literature concerning a priori bounds are immediate consequences of this estimate. In particular, among other results, we establish a priori $L^{\infty }$ bound in dimensions n ≤ 9, under the extra assumption that $\limsup _{t\rightarrow \infty }\frac {f(t)f^{\prime \prime }(t)}{f^{\prime }(t)^{2}}< \frac {2}{9-2\sqrt {14}}\cong 1.318$ . Also, we establish a priori $L^{\infty }$ bound when n ≤ 5 under the very weak assumption that, for some e > 0, $\liminf _{t\rightarrow \infty }\frac {(tf(t))^{2-\epsilon }}{f^{\prime }(t)}>0$ or $\liminf _{t\rightarrow \infty }\frac {t^{2}f(t)f^{\prime \prime }(t)}{f^{\prime }(t)^{\frac {3}{2}+\epsilon }}>0$ .